Factor the following expression: $-7$ $x^2$ $-10$ $x$ $-3$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-7)}{(-3)} &=& 21 \\ {a} + {b} &=& & & {-10} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $21$ and add them together. The factors that add up to ${-10}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-3}$ and ${b}$ is ${-7}$ $ \begin{eqnarray} {ab} &=& ({-3})({-7}) &=& 21 \\ {a} + {b} &=& {-3} + {-7} &=& -10 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-7}x^2 {-3}x {-7}x {-3} $ Group the terms so that there is a common factor in each group: $ ({-7}x^2 {-3}x) + ({-7}x {-3}) $ Factor out the common factors: $ x(-7x - 3) + 1(-7x - 3) $ Notice how $(-7x - 3)$ has become a common factor. Factor this out to find the answer. $(-7x - 3)(x + 1)$